Math students who begin their journey into absolute value usually evaluate expressions with absolute value as “always positive.” That is until they encounter the absolute value of zero, and then their answers become “always positive or zero.”

The formal definition of absolute value is |x| = x if x ≥ 0 or –x if x < 0. The negative x confuses students, and they never quite understand that it is the absolute value that is always positive or zero. Unless this misunderstanding is corrected, the situation becomes more problematic when solving inequalities that involve absolute value, which can lead to unhappy teachers and muddled students who usually conclude, “we don’t like math.”

In our Elevated Math lessons we make it clear that absolute value is distance, and distance is always positive or zero. We begin in lesson M3.1 with instruction on negative numbers followed by problems, and then we introduce the concept of opposite numbers before explaining absolute value:

Here is how we do it:

Lesson M3.1 continues with an explanation of integers, how they relate on a number line, and how to order integers.

In our algebra lessons absolute value is defined again in lesson A6.1 followed by the solving of basic absolute value equations. Here is a portion of that lesson:

The lesson continues with problems and more instruction that includes solving absolute value equations with one and no solutions.

A subsequent lesson, A6.2, gives instruction and solves problems for more advanced absolute value equations, which include two-step linear equations.

When the student reaches lessons A6.3 “Inequalities with ‘Absolute Value is Less Than’” and A6.4 “Inequalities with ‘Absolute Value is Greater Than,’” they now have a firm grasp of the absolute value concept.

In the last lesson of this module, A6.5 “Problems with Absolute Value Equations,” modeling scenarios are modeled using an absolute value equation or inequality. Using all of these Elevated Math lessons can lead to happy teachers and successful, confident students who conclude, “we like math very much!”